Responsible party: vinc0053, Green Suit

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Main class wiki page: home

Please try to include the following

• main points understood, and expand them - what is your understanding of what the points were.
  • expand these points by including many of the details the class discussed.
• main points which are not clear. - describe what you have understood and what the remain questions surrounding the point(s).
  • Other classmates can step in and clarify the points, and expand them.
• How the main points fit with the big picture of QM. Or what is not clear about how today's points fit in in a big picture.
• wonderful tricks which were used in the lecture.
  

notation: |1/2 -1/2>
What is the big picture? Why do we do the math?
<math>\chi_+ = \begin{pmatrix} 1
0\end{pmatrix}</math>
<math>\chi_- = \begin{pmatrix} 0
1\end{pmatrix}</math>
<math>|“m”| =< l</math>
S_e,p,n,q,nu = 1/2 is an intrinsic property that doesn't change. You can just say S_z,m_s,m = +/- 1/2 <math>\hbar</math> ←(the <math>\hbar</math> is really supposed to be there, but it is usually left out of the notation).
S² = <math>\frac{1}{2}(\frac{1}{2}+1)\hbar</math>². Compare to the shallow well with only 2 states.
<math>\psi = c_1\psi_1 + c_2\psi_2</math> ⇒ <math>\begin{pmatrix} c_1
c_2\end{pmatrix}</math>, then <math>\begin{pmatrix} 1
0\end{pmatrix}</math> is <math>\psi_1</math> the ground state and <math>\begin{pmatrix} 0
1\end{pmatrix}</math> is the first excited state <math>\psi_2</math>.
When you have spin momentum it creates a magnetic dipole, <math>\psi_1</math> and <math>\psi_2</math> are degenerate states when without a magnetic field. Apply magnetic field and <math>\psi_1</math>, <math>\psi_2</math> take different energies.

Q: How do we know they are degenerate without a magnetic field?
A: <math>\psi = \psi_nlm</math> s_z +/- <math>\frac{1}{2}</math>. When it decays there's emission of photon with energy equal to the difference. The spin decay +/- <math>\frac{1}{2}</math> → +/- <math>\frac{1}{2}</math> makes no difference of photon energy without a magnetic field, but with it, experiments started showing a difference. Two peaks on a plot of measurements (which collapse to one peak without a magnetic field).
We could use convention <math>\chi_+</math>^x = <math>\begin{pmatrix} 1
0\end{pmatrix}</math> <math>\chi_-</math>^x = <math>\begin{pmatrix} 0
1\end{pmatrix}</math> for spin up and spin down, but its confusing. If the z-component is discrete, not continuous, then it snaps into allowed spots (up or down). There can only be an even number of allowed spots, why is that?
S²,s_z,s_x,s_y (sometimes the x and y components) like a microwave has a magnetic field that lines up hydrogen in H2O, then microwaves flip the spin over.
S² = <math>\frac{3}{4}\hbar</math>²<math>\begin{pmatrix} 1 0
0 -1\end{pmatrix}</math> ⇒ <math>\frac{3}{4}\hbar</math>²<math>\begin{pmatrix} 1 0
0 -1\end{pmatrix}\begin{pmatrix} 1
0\end{pmatrix} = \frac{3}{4}\hbar</math>²<math>\begin{pmatrix} 1
0\end{pmatrix}</math>

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